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1: 3.5 Quadrature
§3.5(v) Gauss Quadrature
Gauss–Laguerre Formula
§3.5(viii) Complex Gauss Quadrature
a complex Gauss quadrature formula is available. …
2: 35.10 Methods of Computation
Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on 𝐎 ( m ) applied to a generalization of the integral (35.5.8). …
3: Bibliography G
  • C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.
  • W. Gautschi (2016) Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.
  • G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.
  • 4: Bibliography I
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • A. Iserles, S. P. Nørsett, and S. Olver (2006) Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.
  • K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida (1991) From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.
  • 5: 15.19 Methods of Computation
    The Gauss series (15.2.1) converges for | z | < 1 . … Large values of | a | or | b | , for example, delay convergence of the Gauss series, and may also lead to severe cancellation. For fast computation of F ( a , b ; c ; z ) with a , b and c complex, and with application to Pöschl–Teller–Ginocchio potential wave functions, see Michel and Stoitsov (2008). … Gauss quadrature approximations are discussed in Gautschi (2002b). … For example, in the half-plane z 1 2 we can use (15.12.2) or (15.12.3) to compute F ( a , b ; c + N + 1 ; z ) and F ( a , b ; c + N ; z ) , where N is a large positive integer, and then apply (15.5.18) in the backward direction. …
    6: 20.11 Generalizations and Analogs
    §20.11(i) Gauss Sum
    For relatively prime integers m , n with n > 0 and m n even, the Gauss sum G ( m , n ) is defined by … … Similar identities can be constructed for F 1 2 ( 1 3 , 2 3 ; 1 ; k 2 ) , F 1 2 ( 1 4 , 3 4 ; 1 ; k 2 ) , and F 1 2 ( 1 6 , 5 6 ; 1 ; k 2 ) . …
    7: 9.17 Methods of Computation
    For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). … The second method is to apply generalized Gauss–Laguerre quadrature3.5(v)) to the integral (9.5.8). … For quadrature methods for Scorer functions see Gil et al. (2001), Lee (1980), and Gordon (1970, Appendix A); but see also Gautschi (1983). …
    8: 15.5 Derivatives and Contiguous Functions
    The six functions F ( a ± 1 , b ; c ; z ) , F ( a , b ± 1 ; c ; z ) , F ( a , b ; c ± 1 ; z ) are said to be contiguous to F ( a , b ; c ; z ) .
    15.5.11 ( c a ) F ( a 1 , b ; c ; z ) + ( 2 a c + ( b a ) z ) F ( a , b ; c ; z ) + a ( z 1 ) F ( a + 1 , b ; c ; z ) = 0 ,
    15.5.12 ( b a ) F ( a , b ; c ; z ) + a F ( a + 1 , b ; c ; z ) b F ( a , b + 1 ; c ; z ) = 0 ,
    By repeated applications of (15.5.11)–(15.5.18) any function F ( a + k , b + ; c + m ; z ) , in which k , , m are integers, can be expressed as a linear combination of F ( a , b ; c ; z ) and any one of its contiguous functions, with coefficients that are rational functions of a , b , c , and z . …
    15.5.20 z ( 1 z ) ( d F ( a , b ; c ; z ) / d z ) = ( c a ) F ( a 1 , b ; c ; z ) + ( a c + b z ) F ( a , b ; c ; z ) = ( c b ) F ( a , b 1 ; c ; z ) + ( b c + a z ) F ( a , b ; c ; z ) ,
    9: 6.18 Methods of Computation
    Quadrature of the integral representations is another effective method. For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968). … Power series, asymptotic expansions, and quadrature can also be used to compute the functions f ( z ) and g ( z ) . …
    10: 15.10 Hypergeometric Differential Equation
    f 1 ( z ) = F ( a , b c ; z ) ,
    f 1 ( z ) = F ( a , b a + b + 1 c ; 1 z ) ,
    (b) If c equals n = 1 , 2 , 3 , , and a 1 , 2 , , n 1 , then fundamental solutions in the neighborhood of z = 0 are given by F ( a , b ; n ; z ) and …
    15.10.11 w 1 ( z ) = F ( a , b c ; z ) = ( 1 z ) c a b F ( c a , c b c ; z ) = ( 1 z ) a F ( a , c b c ; z z 1 ) = ( 1 z ) b F ( c a , b c ; z z 1 ) .
    The ( 6 3 ) = 20 connection formulas for the principal branches of Kummer’s solutions are: …